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Theorem sprel 43931
Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
sprel (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6694 . 2 (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2 sprvalpw 43927 . . . 4 (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
32eleq2d 2901 . . 3 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}}))
4 eqeq1 2828 . . . . . 6 (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
542rexbidv 3292 . . . . 5 (𝑝 = 𝑋 → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
65elrab 3666 . . . 4 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
76simprbi 500 . . 3 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
83, 7syl6bi 256 . 2 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
91, 8mpcom 38 1 (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  Vcvv 3480  𝒫 cpw 4522  {cpr 4552  cfv 6343  Pairscspr 43924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-spr 43925
This theorem is referenced by:  prssspr  43932  prsprel  43934  reupr  43969
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